EN
Let G be an infinite, compact abelian group and let Λ be a subset of its dual group Γ. We study the question which spaces of the form $C_{Λ}(G)$ or $L¹_{Λ}(G)$ and which quotients of the form $C(G)/C_{Λ}(G)$ or $L¹(G)/L¹_{Λ}(G)$ have the Daugavet property.
We show that $C_{Λ}(G)$ is a rich subspace of C(G) if and only if $Γ∖ Λ^{-1}$ is a semi-Riesz set. If $L¹_{Λ}(G)$ is a rich subspace of L¹(G), then $C_{Λ}(G)$ is a rich subspace of C(G) as well. Concerning quotients, we prove that $C(G)/C_{Λ}(G)$ has the Daugavet property if Λ is a Rosenthal set, and that $L¹_{Λ}(G)$ is a poor subspace of L¹(G) if Λ is a nicely placed Riesz set.